Mod-3-Cohomology of group number 9 of order 36

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General information on the group

  • The group order factors as 22 · 32.
  • It is non-abelian.
  • It has 3-Rank 2.
  • The centre of a Sylow 3-subgroup has rank 2.
  • Its Sylow 3-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.


Structure of the cohomology ring

The computation was based on 3 stability conditions for H*(SmallGroup(9,2); GF(3)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    1  −  2·t  +  4·t2  −  4·t3  +  4·t4  −  4·t5  +  4·t6  −  2·t7  +  t8

    ( − 1  +  t)2 · (1  +  t2)2 · (1  +  t4)
  • The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

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Ring generators

The cohomology ring has 8 minimal generators of maximal degree 8:

  1. a_2_0, a nilpotent element of degree 2
  2. a_3_1, a nilpotent element of degree 3
  3. a_3_0, a nilpotent element of degree 3
  4. c_4_0, a Duflot element of degree 4
  5. a_7_3, a nilpotent element of degree 7
  6. a_7_2, a nilpotent element of degree 7
  7. c_8_2, a Duflot element of degree 8
  8. c_8_1, a Duflot element of degree 8

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Ring relations

There are 4 "obvious" relations:
   a_3_02, a_3_12, a_7_22, a_7_32

Apart from that, there are 16 minimal relations of maximal degree 16:

  1. a_2_02
  2. a_2_0·a_3_0
  3. a_2_0·a_3_1
  4. a_3_0·a_3_1 + a_2_0·c_4_0
  5. a_2_0·a_7_2
  6. a_2_0·a_7_3
  7. a_3_0·a_7_2 + a_2_0·c_8_2
  8. a_3_0·a_7_3 + a_2_0·c_8_1
  9. a_3_1·a_7_2 − a_2_0·c_8_1
  10. a_3_1·a_7_3 + a_2_0·c_8_2
  11. c_8_2·a_3_0 − c_8_1·a_3_1 + c_4_0·a_7_3
  12. c_8_2·a_3_1 + c_8_1·a_3_0 − c_4_0·a_7_2
  13. a_7_2·a_7_3 + a_2_0·c_4_0·c_8_2
  14. c_8_2·a_7_2 + c_8_1·a_7_3 + c_4_0·c_8_1·a_3_0 − c_4_02·a_7_2
  15. c_8_2·a_7_3 − c_8_1·a_7_2 + c_4_0·c_8_1·a_3_1 − c_4_02·a_7_3
  16. c_8_22 + c_8_12 − c_4_02·c_8_2


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Data used for the Hilbert-Poincaré test

  • We proved completion in degree 16 using the Hilbert-Poincaré criterion.
  • The completion test was perfect: It applied in the last degree in which a generator or relation was found.
  • The following is a filter regular homogeneous system of parameters:
    1. c_4_0, an element of degree 4
    2. c_8_2, an element of degree 8
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 10].


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Restriction maps

Expressing the generators as elements of H*(SmallGroup(9,2); GF(3))

  1. a_2_0a_1_0·a_1_1
  2. a_3_1c_2_2·a_1_0 − c_2_1·a_1_1
  3. a_3_0c_2_2·a_1_1 + c_2_1·a_1_0
  4. c_4_0c_2_22 + c_2_12
  5. a_7_3c_2_1·c_2_22·a_1_0 + c_2_12·c_2_2·a_1_1
  6. a_7_2c_2_1·c_2_22·a_1_1 − c_2_12·c_2_2·a_1_0
  7. c_8_2c_2_12·c_2_22
  8. c_8_1c_2_1·c_2_23 − c_2_13·c_2_2

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2

  1. a_2_0a_1_0·a_1_1, an element of degree 2
  2. a_3_1c_2_2·a_1_0 − c_2_1·a_1_1, an element of degree 3
  3. a_3_0c_2_2·a_1_1 + c_2_1·a_1_0, an element of degree 3
  4. c_4_0c_2_22 + c_2_12, an element of degree 4
  5. a_7_3c_2_1·c_2_22·a_1_0 + c_2_12·c_2_2·a_1_1, an element of degree 7
  6. a_7_2c_2_1·c_2_22·a_1_1 − c_2_12·c_2_2·a_1_0, an element of degree 7
  7. c_8_2c_2_12·c_2_22, an element of degree 8
  8. c_8_1c_2_1·c_2_23 − c_2_13·c_2_2, an element of degree 8


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Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2



Last change: 14.12.2010